Ordering properties of convolutions of heterogeneous Erlang and Pascal random variables

In this paper, we study ordering properties of convolutions of heterogeneous Erlang and Pascal random variables in terms of the majorization order [pp-larger order, reciprocal majorization order] of parameter vectors and the likelihood ratio order [hazard rate order, mean residual life order]. We establish, among other things, that weak majorization order [pp-larger order, reciprocal majorization order] between scale parameter vectors and majorization order between shape parameter vectors imply likelihood ratio order [hazard rate order, mean residual life order] between convolutions of two heterogeneous Erlang or Pascal sets of variables. These results strengthen and generalize some of the results known from the literature.